Median (in statistics)

2020 Mathematics Subject Classification: Primary: 60-01 Secondary: 62-01 [MSN][ZBL]

One of the numerical characteristics of probability distributions, a particular case of a quantile. For a real-valued random variable $X$ with distribution function $F$, a median is defined as a number $m$ such that $F(m)\leq1/2$ and $F(m+0)\geq1/2$. Every random variable has at least one median. If $F(x)=1/2$ for all $x$ in a closed interval, then every point of this interval is a median. If $F$ is a strictly-monotone function, then the median is unique. In the symmetric case, if the median is unique, it is identical with the mathematical expectation, provided that the latter exists. The fact that a median always exists is used for centering random variables (see, for instance, Lévy inequality). In mathematical statistics, to estimate the median of a distribution in terms of independent results of observations $X_1,\dots,X_n$ one uses a so-called sample median — a median of the corresponding order statistics (cf. Order statistic) $X_{(1)},\dots,X_{(n)}$, that is, of the quantity $X_{(k+1)}$ if $n=2k+1$ is odd, and of $[X_{(k)}+X_{(k+1)}]/2$ if $n=2k$ is even.

References

[L]	M. Loève, "Probability theory" , Springer (1977) MR0651017 MR0651018 Zbl 0359.60001
[C]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) MR0016588 Zbl 0063.01014